Start by adding the 1/4 cup of lukewarm water, active dry yeast and 1 teaspoon of sugar to a small glass measuring cup, whisking together 'til well dissolved. Set aside in a warm place so that the yeast can "proof;" After a few minutes, it should become foamy and roughly double in volume.
In a medium mixing bowl, whisk together the all-purpose flour and salt. Set aside.
In a another large mixing bowl or in the bowl of an electric stand mixer, cream the rest of the sugar with the softened butter, 'til well combined.
Beat in each egg, one at a time, 'til incorporated; be sure to stop and scrape up the bottom of the bowl a couple of times, beating in any of the butter/sugar bits that were left behind.
Once your yeast has proofed, go ahead and add it to the egg mixture, and then add the warm buttermilk. Whisk everything together.
Finally, add the dry mixture to the wet mixture, and use a large wooden spoon or a dough scraper to mix everything together for a few minutes, 'til everything is well combined and you have a soft, somewhat sticky dough. If you're using an electric mixer, you'll want to use the dough hook for this step.
Cover your mixing bowl with plastic wrap, and let the dough rise in a warm place for about 1 to 1 1/2 hours**, or 'til roughly doubled in volume.
Once your donut dough has roughly doubled in volume, use a dough scraper or a scraper spatula to turn it out onto a floured work surface. Sprinkle a little flour on top, and roll it out with a rolling pin to about 1/2" thickness, adding more flour as necessary.
Use a donut cutter to cut out your donuts. If you don't want to keep the donut holes, you can re-roll them back out with the scraps, and then cut out more full-size donuts.
Place each donut onto a well floured piece of parchment paper. Cover with a lightweight towel, and let rise for another 1 to 1 1/2 hours, or 'til roughly doubled in size.